Algebra 101 - Unit 12

Lesson 71: Circles

A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius (r). The longest distance across the circle passing through the center is the diameter (d = 2r).

Equation of a Circle

- Standard form: (x − h)² + (y − k)² = r², where (h, k) is the center. - General form: x² + y² + Dx + Ey + F = 0

Examples

Example 1: Find the center and radius of (x − 3)² + (y + 2)² = 25

Solution: Center = (3, −2), Radius = √25 = 5

Example 2: Write the equation of a circle with center (−1, 4) and radius 6

Solution: (x + 1)² + (y − 4)² = 36

Example 3: Convert x² + y² − 4x + 6y − 12 = 0 to standard form

Solution: Complete the square: (x² − 4x + 4) + (y² + 6y + 9) = 12 + 4 + 9 → (x − 2)² + (y + 3)² = 25 Center = (2, −3), Radius = 5

Practice Problems

1. Find the center and radius of (x + 5)² + (y − 1)² = 49
2. Write the equation of a circle with center (2, −3) and radius 7
3. Convert x² + y² + 6x − 8y + 9 = 0 to standard form and find the center and radius
4. Determine if the point (1, 2) lies on the circle (x − 1)² + (y − 3)² = 10
5. Find the diameter of a circle with equation (x − 4)² + (y + 5)² = 16